ST Assignement 2:¶

Hocine Samy¶
Bazaz Jazayeri Samuel¶

Setup¶

In [1]:
############
# Packages #
############
import time as t
import numpy as np
import pandas as pd

import matplotlib.pyplot as plt
import plotly.express as px
import seaborn as sns

from numba import jit
from tqdm.notebook import tqdm
from scipy.signal import periodogram, fftconvolve
from sklearn.linear_model import LinearRegression
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf
import plotly.io as pio
pio.renderers.default='notebook'
###################
# hyperparametres #
###################

C:\Users\samud\anaconda3\lib\site-packages\cupy\_environment.py:447: UserWarning: 
--------------------------------------------------------------------------------

  CuPy may not function correctly because multiple CuPy packages are installed
  in your environment:

    cupy-cuda101, cupy-cuda12x

  Follow these steps to resolve this issue:

    1. For all packages listed above, run the following command to remove all
       existing CuPy installations:

         $ pip uninstall <package_name>

      If you previously installed CuPy via conda, also run the following:

         $ conda uninstall cupy

    2. Install the appropriate CuPy package.
       Refer to the Installation Guide for detailed instructions.

         https://docs.cupy.dev/en/stable/install.html

--------------------------------------------------------------------------------

  warnings.warn(f'''

Utility functions

Question 3¶

Firstly let's generate an AR2.

In [2]:
def gen_ar2(phi1,phi2, n_step, y0=None, y1=None):
    signal = np.random.normal(1, 1, n_step)
    if y0:
        signal[0] = y0
    if y1:
        signal[1] = y1
        
    for i in range(2,n_step):
        signal[i] = signal[i]+ phi1*signal[i-1]+phi2*signal[i-2]
    return signal

def plot_signal(signal, title = "<b>Signal</b>"):
    fig = px.line(signal, title = title, template="plotly_dark")
    fig.show()
In [3]:
def get_the_ar2(nstep, y0=None, y1=None):
    phi1 = (-1/1.05)
    phi2 = (-1/(1.05**2))
    signal = gen_ar2(phi1,phi2,nstep,y0=y0, y1=y1)
    plot_signal(signal, title = f"<b>AR 2 </b> phi1={np.round(phi1, 2)} phi2={np.round(phi2, 2)}<br> y0={y0} y1={y1}")
    return signal
In [4]:
ar2_noinit = get_the_ar2(2000, y0=None, y1=None)
In [5]:
ar2_init = get_the_ar2(2000, y0=30, y1=20)

The previous graphs show that the influence of initialization is limited, due to the exponential decrease in autocorrelation (see below).

In [6]:
plt.style.use('dark_background')
plot_acf(ar2_noinit, title = "Autocorrelation AR2 random initialisation")
plt.show()
No description has been provided for this image
In [7]:
plt.style.use('dark_background')
plot_acf(ar2_init, title = "Autocorrelation AR2 y0=30 y1=20")
plt.show()
No description has been provided for this image

Moreover, the autocorrelation oscillates because the roots are complex.

Now let's compute the periodogram with fs=1Hz.

In [8]:
def plot_periodogram(signal, title = "<b>Periodogram</b>", fs=1):
    lst_f, perio = periodogram(signal, fs)
    df = pd.DataFrame({"frequency":lst_f, "power spectrum estimation":perio})
    fig = px.line(df, x="frequency", y="power spectrum estimation", title = title, template="plotly_dark")
    fig.show()
In [9]:
plot_periodogram(ar2_noinit, title = "<b>Periodogram AR2</b> random initialization")
In [10]:
plot_periodogram(ar2_init, title = "<b>Periodogram AR2</b> y0 = 30 y1 = 20")

Question 4¶

The signal to transform is the following:

In [11]:
signal_question_4 = np.array([-0.4660063139418452, 0.15203881196316202, 0.15819979628299907, 0.11085881533376842, 0.3702551247873387, -0.14764939761668475, 0.0005242759871536551, 0.02359456400585025, 0.1596483710856063, -0.323761952319536, -0.1475686421106783, -0.28142045506984575, -0.814422646968135, -0.16391894343863994, 0.23146644702935346, -0.29047335918346834, 0.06401571071714088, 0.24103338271568847, -0.16018841576610637, -0.20228284480286607, -0.21744836626816816, -0.2070277709613954, 0.8640548582335627, 0.3469532848247603, -0.16319241626905354, -0.015561809939087123, -0.7927845042595223, -0.5898984115750532, -0.2939025080841799, 0.2912999336934706, 0.49592358486052546, 0.02746759984182684, 0.07927812951132095, -0.41453289673010874, -1.037027302591971, -0.2729078256384515, -0.07067873207917819, -0.12004978967279764, 0.5055075840060094, 0.7214288056485215, -0.04984298866407566, 0.17843881471342735, -0.4537829723639949, -0.012521870053272766, -0.4276141859259953, 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0.18150667762354267, 0.030810457465984764, 0.2201294801866378, -0.28730950958835044, 0.025164216241136864, -0.0789787768657246, -0.2855774529280901, 0.06411300860940417, 0.25417300501353796, 0.12892906199646104, 0.44033382533626064, 0.3730265523640188, -0.2402623013666117, -0.0021160570615325613, -0.4999191070587656, 0.6427493495038137, 0.03503942104350481, 0.512852960590843, 0.20399742880003213, -0.290882569087827, -0.15926032809949825, 0.5838721695495923, 0.02523513764982602, 0.03279531363043672, -0.17862377080145975, 0.43710046624117155, 0.34551605776770417, 0.3880685167261527, 0.6233962791584153, 0.04548336914124056, -0.31920854270338894, -0.3028516948418159, -0.31140790911784944, 0.21279286106298048, -0.07006637673550706, 0.24573486282034165, 0.31241177173589685, 0.37407456947162016, 0.16996621448596205, 0.226086453570638, -0.37367146579086935, -0.16523919383143293, -0.34993597029424456, -0.08673855859425618, 0.16911521679183308, -0.5480489939053387, 0.12055628273790664, -0.570749542755565, 0.18803244242690895, -0.15313153614180974, -0.17006829227592435, 0.35951786046477713, -0.330287701076994, -0.6803256387755492, -0.11683140768062408, 0.3684939414956032, -0.7409107019900307, 0.013518419792164805, -0.19396913986959183, 0.04476633340131024, -0.12719167154625982])
In [12]:
plot_signal(signal_question_4, title = "<b>Signal to transform with MDCT</b>")
In [13]:
def plot_signals(df, title = "<b>Signal</b>"):
    fig = px.line(df, title = title, template="plotly_dark", color_discrete_sequence =sns.color_palette("Spectral", df.shape[1]).as_hex())
    fig.show()

We build here the different patterns used in the construction of the database.

In [14]:
@jit(nopython=True)
def get_phi_l_k(l,k):
    u = np.arange(2*l)
    return np.sin(np.pi/(2*l)*(u+0.5))*np.sqrt(2/l)*np.cos(np.pi/l*(u+(l+1)/2)*(k+0.5))
In [15]:
for l in [32,64,128,256]:
    plot_signals(pd.DataFrame({f"l : {l}  k : {k}": get_phi_l_k(l,k) for k in range(0,l,l//4)}), title = f"<b>phi_{l}_k MDCT</b>")

Now we use translations in order to have the vectors of our dictionary.

In [16]:
@jit(nopython=True)
def get_dico_vec_l_k_i(l,k,i, signal_length = 2048):
    pattern_length = 2*l
    dico_vec = np.zeros(signal_length)
    dico_vec[i:i+pattern_length] = get_phi_l_k(l,k)
    return dico_vec
In [17]:
for l in [32,64]:
    plot_signals(pd.DataFrame({f"l : {l}  k : {l//2} i : {i}": get_dico_vec_l_k_i(l,l//16,i) for i in range(100,2000, 200)}), title = f"<b>phi_{l}_{l//2} MDCT</b>")

As our models are theoretically normalized, so are the preceding vectors. Let's check, however, that the numerical error is not too large.

In [18]:
log_norm_abserrors = []
for l in [32, 64, 128, 256, 512, 1024]:
    for k in range(l):
        vec_norm = np.linalg.norm(get_phi_l_k(l,k))
        if vec_norm !=1:
            log_norm_abserrors.append( np.log10(np.abs(vec_norm-1)))
fig = px.scatter(log_norm_abserrors, title = "exponents of absolute errors on the norm")
fig.show()

So our dictionary can be considered as a normalized family. However, this family is not orthogonal:

In [19]:
np.dot(get_dico_vec_l_k_i(64,8,0), get_dico_vec_l_k_i(128,16,0))
Out[19]:
0.3531756524052614

This is why a heuristic is needed to find the best sparse decomposition of our signal instead of a classical PCA. Hence the following algorithm: OMP. The idea is to iteratively select a new vector from the base as being the most correlated with the residual from the previous step, then calculate a linear regression of the signal from all previously selected vectors to update the coordinates.

Instead of calculating all the shifted patterns, let's calculate the scalar product for all the shifts using convolution and the FFT. By doing so we need to save only the patterns.

In [20]:
@jit(nopython=True)
def get_all_patterns(lst_l = np.array([32, 64, 128, 256, 512, 1024])):
    patterns = {}
    for l in lst_l:
        patterns_l = np.zeros((l, 2*l))
        for k in np.arange(l):
#         for k in np.arange(l//2):
            patterns_l[k,:] = get_phi_l_k(l,k)
        patterns[l] = patterns_l.copy()
    return patterns
In [21]:
patterns = get_all_patterns()
In [22]:
plot_signals(patterns[32].T, "All patterns for l=32")

C:\Users\samud\anaconda3\lib\site-packages\numba\typed\typeddict.py:38: NumbaTypeSafetyWarning:

unsafe cast from int64 to int32. Precision may be lost.

Now let's test the computation of scalar product without and with convolution.

In [23]:
def get_ps_profiles(signals, patterns_l):
    len_pat =  patterns_l.shape[1]
    len_signals = signals.shape[1]
    res = np.zeros((patterns_l.shape[0], len_signals-len_pat+1))
    for i in range(len_signals-len_pat+1):
        res[:,i] = np.sum(patterns_l*signals[:,i:i+len_pat], axis = 1)
    return res

def get_ps_profiles_conv(signals, patterns_l):
    len_pat =  patterns_l.shape[1]
    convol =  fftconvolve(signals, np.flip(patterns_l, axis=1), axes = 1)
    if len_pat ==1:
        return convol
    return convol[:, len_pat-1:1-len_pat]
In [24]:
# test 
signals = np.tile(signal_question_4, 32).reshape((32,2048))
t1 = t.time()
mat_ps = get_ps_profiles(signals, patterns[32])
t2 = t.time()
mat_ps_conv = get_ps_profiles_conv(signals, patterns[32])
t3 = t.time()
In [25]:
plot_signals(mat_ps.T, f"Scalar products of the signals with patterns for l=32 | {np.round(t2-t1,3)}s")
In [26]:
plot_signals(mat_ps_conv.T, f"Scalar products using convolution for l=32 | {np.round(t3-t2,3)}s")
In [27]:
def get_argmax_and_max_from_mat(mat):
    argmax =  np.unravel_index(np.argmax(mat, axis=None), mat.shape)
    return argmax, mat[argmax]
In [28]:
print(get_argmax_and_max_from_mat(np.abs(mat_ps)))

((0, 1065), 10.140575598376582)
In [29]:
def get_argmax_abs_ps_and_add_to_selection(patterns, residual, dct_selection):
    residual_repeated = np.tile(residual.copy(), 2048).reshape((2048,2048))
    l_max = None
    k_i_max =None
    ps_max = None
    
    for l, patterns_l in patterns.items():
        mat = get_ps_profiles_conv(residual_repeated[:l,:], patterns_l)
        mat = np.abs(mat)
        
        for k,i in dct_selection[l]:
            mat[k,i] = 0
        argmax_l, ps_max_l =  get_argmax_and_max_from_mat(mat)
        
        if l_max:
            if ps_max_l>ps_max:
                l_max = l
                k_i_max = argmax_l
                ps_max = ps_max_l
        else:
            l_max = l
            k_i_max = argmax_l
            ps_max = ps_max_l
    
    dct_selection[l_max].append(k_i_max)
In [30]:
dct_selection = { 32:[]} 
for k in range(3):
    get_argmax_abs_ps_and_add_to_selection({32:patterns[32]}, signal_question_4, dct_selection)
print(dct_selection)

{32: [(0, 1065), (0, 1066), (0, 1064)]}
In [31]:
def mat_vec_from_dct_selection(dct_selection):
    lst_vec = []
    for l, lst_ki in dct_selection.items():
        for k,i in lst_ki:
            lst_vec.append(get_dico_vec_l_k_i(l,k,i, signal_length = 2048))
    return np.array(lst_vec).T
In [32]:
plot_signals(mat_vec_from_dct_selection(dct_selection), "first vectors selected if only l=32")
In [33]:
def orthogonal_matching_pursuit(signal, ncomp=10, lst_l = [32, 64, 128, 256, 512, 1024], debug = False):
    #ncomp : number of non-null coefficients
    # Initializations 
    coordinates = np.zeros(ncomp)
    residual = signal.copy()
    lst_norm = []       
    patterns = get_all_patterns( np.array(lst_l) )
    dct_selection = { l:[] for l in lst_l} 
                           
    for i in tqdm(range(ncomp)):
        get_argmax_abs_ps_and_add_to_selection(patterns, residual, dct_selection)
        mat_selection = mat_vec_from_dct_selection(dct_selection).copy()
        reg_lin = LinearRegression().fit(mat_selection, signal)
        coordinates = reg_lin.coef_
        approx = reg_lin.predict(mat_selection)
        residual = signal - approx
        residual_norm = np.linalg.norm(residual)
         
        if debug:
            print(dct_selection)
            print(coordinates)
            print(residual_norm)
  
        lst_norm.append(residual_norm.copy())

    return coordinates, approx, lst_norm
In [34]:
coordinates_40, approx_40, lst_norm_40 = orthogonal_matching_pursuit(signal_question_4, ncomp=40, debug = False )

  0%|          | 0/40 [00:00<?, ?it/s]
In [35]:
plot_signal(lst_norm_40, "Norm of the residuals OMP")
In [36]:
coordinates_10, approx_10, lst_norm_10 = orthogonal_matching_pursuit(signal_question_4, ncomp=10, debug = False )

  0%|          | 0/10 [00:00<?, ?it/s]
In [37]:
plot_signals(pd.DataFrame({"signal_question_4":signal_question_4, "10 atomes":approx_10, "40 atomes":approx_40}), "Orthogonal Matching Pursuit")
In [ ]: